To quantify properties of parts (cast or otherwise) one may need to design a test specific to the part. Usual questions answered by this tend to be along the lines of: "Will my part be able to handle what it needs to?" The amount of testing is usually narrowed down via a theoretical model. For example one might attach strain gages at specific locations based on a model and subject the part to loads gathering information to support the model. Other tests may include laser vibrometry, cyclic loading and testing to failure. It all depends on the function of the part, and will likely involve multiple tests and statistics to prove that the results apply to similar parts made via your process. When in doubt, the test becomes use of the part as it is meant to be used, for the life of the part while measuring for what constitutes failure of the part to perform its function. The need for enough historical data to be able to narrow down required testing for each part variant can make entry into industry difficult; standardized parts are a summarization of such results intended to make it easier to produce specific parts.
To quantify properties of a cast material, one casts (and then machines) to standard material testing shapes. Testing the specimens then goes toward generating material properties. Continuing to generate and test them with batches of parts produced can support (along with additional statistical data proving correlations) the claim that material used in one batch is similar to that used in another. The results can be used to model properties of a part, but such models then need to be validated.
There are also cases where a standardized specimen can be machined out of a specific locations on a part. Specimens (and test results) are categorized by these specific locations since the conditions in the manufacturing process can vary by location. The results provide support to models of the process that made the specimens and can often be applied to infer properties at other locations on the part (helping model the part), contributing to quantifying both material and part properties.
Ultimately it comes down to mathematical models, test results that support those models, and people with enough knowledge to design models to accurately correspond to reality and tests to cover your part's function.